Even though the $k$-NN density estimator is already adaptive to the data, there is still a global value $k$ that controls the amount of smoothing in the entire space. Is using multiple values of $k$ in different subsets of the data wrong? My intuition says that at the end of the day, the estimates mean "number of points per volume of space", and it wouldn't be a problem, but I can't seem to find out if that's actually OK, or if this has been done in the past.
I can see how that would not be a problem if used along a kernel function that would generate a proper PDF at the end. Not sure if this has any serious theoretical implications on the $k$-NN estimator.
Thank you!